# Evaluate the following iterated integrals. int_1^2int_0^1(3x^2+4y^3)dydx

Question
Integrals
Evaluate the following iterated integrals.
$$\int_1^2\int_0^1(3x^2+4y^3)dydx$$

2021-01-11
To evaluate the integral: $$\int_1^2\int_0^1(3x^2+4y^3)dydx$$
Solution:
When we integrate with respect to one variable then other will be kept as constant.
Evaluating the integral.
$$\int_1^2\int_0^1(3x^2+4y^3)dy=\int_1^2[\int_0^1(3x^2+4y^3)dy]dx$$
$$=\int_1^2 [(3x^2y+4\cdot\frac{y^4}{4})_0^1]dx$$
$$=\int_1^2 [3x^2\cdot 1+1^4-0]dx$$
$$=\int_1^2 (3x^2+1)dx$$
$$=[3\cdot\frac{x^3}{3}+x]_1^2$$
$$=[x^3+x]_1^2$$
$$=[(2^3+2)-(1^3+1)]$$
$$=[12-2]$$
$$=10$$
$$\text{Hence, required answer is 10}$$

### Relevant Questions

Evaluate the following iterated integrals.
$$\int_0^2\int_0^1 4xy\ dx\ dy$$
Evaluate the following iterated integrals.
$$\int_0^2\int_0^1\frac{8xy}{1+x^4}dxdy$$
Evaluate the following iterated integrals. $$\int_1^3\int_1^2(y^2+y)dx\ dy$$
Evaluate the following iterated integrals.
$$\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx$$
Evaluate the following iterated integrals.
$$\int_0^3\int_{-2}^1 (2x+3y)dxdy$$
Evaluate the following integrals.
$$\int_1^{e^2}\frac{(\ln x)^5}{x}dx$$
$$\int_{1/8}^1\frac{dx}{x\sqrt{1+x^{2/3}}}$$
$$\int x^2 6^{x^3+8}dx$$
$$\int_{-2}^2\frac{e^{z/2}}{e^{z/2}+1}dz$$
$$\int_{\ln2}^{\ln3}\frac{e^x+e^{-x}}{e^{2x}-2+e^{-2x}}dx$$